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218 8. Tests de copule gaussienne
ν = 3
ν = 5
ν = 8
ν = 20
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.07 0.08 0.07 0.04 0.07
d2 0.03 0.03 0.07 0.04 0.06
d3 0.22 0.17 0.08 0.03 0.01
d4 0.03 0.03 0.08 0.03 0.04
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.46 0.47 0.46 0.52 0.52
d2 0.36 0.34 0.39 0.44 0.43
d3 0.52 0.54 0.47 0.30 0.14
d4 0.37 0.36 0.43 0.45 0.45
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.85 0.86 0.87 0.88 0.89
d2 0.85 0.84 0.86 0.87 0.88
d3 0.91 0.91 0.91 0.81 0.70
d4 0.86 0.85 0.90 0.89 0.90
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.97 0.99 0.97 0.99 0.99
d2 0.99 0.99 0.97 0.99 0.99
d3 0.99 0.99 0.98 0.99 0.97
d4 0.99 0.99 0.98 0.99 0.99
ν = 4
ν = 7
ν = 10
ν = 50
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.28 0.26 0.32 0.30 0.29
d2 0.18 0.17 0.21 0.21 0.24
d3 0.36 0.33 0.26 0.15 0.03
d4 0.18 0.17 0.23 0.21 0.21
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.78 0.81 0.81 0.81 0.86
d2 0.71 0.78 0.76 0.77 0.82
d3 0.80 0.81 0.82 0.73 0.52
d4 0.75 0.81 0.79 0.80 0.83
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.92 0.93 0.96 0.95 0.94
d2 0.93 0.92 0.95 0.96 0.94
d3 0.96 0.96 0.96 0.95 0.88
d4 0.94 0.94 0.96 0.97 0.95
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.99 0.99 0.99 0.99 0.99
d2 0.99 0.99 0.99 0.99 0.99
d3 0.99 0.99 0.99 0.99 0.99
d4 0.99 0.99 0.99 0.99 0.99
Table 1: The values p 95%(ν, ρ) shown in this table give the maximum values that the probability p should
take in order to be able to reject the hypothesis that a Student’s copula with ν degrees and correlation
ρ is undistinguishable from a Gaussian copula at the 95% confidence level. p 95% is the abscissa corresponding
to the ordinate D(p 95%) = 0.95 shown in figures 3 and 4. p is the probability that pairs of
Gaussian random variables with the correlation coefficient ρ have a distance (between the distribution of
z 2 and the theoretical χ 2 distribution) equal to or larger than the corresponding distance obtained for the
Student’s vector time series. A small p corresponds to a clear distinction between Student’s and Gaussian
vectors, as it is improbable that Gaussian vectors exhibit a distance larger than found for the Student’s
vectors. Different values of the number ν of degrees of freedom ranging from ν = 3 to ν = 50 and of
the correlation coefficient ρ = 0.1 to 0.9 are shown. Let us take for instance the example with ν = 4 and
ρ = 0.3. The table indicates that p should be less than about 0.3 (resp. 0.2) according to the distances d1
and d3 (resp. d2 and d4) for being able to distinguish this Student’s copula from the Gaussian copula at
the 95% confidence level. This means that less than 20−30% of Gaussian vectors should have a distance
for their z 2 larger than the one found for the Student’s. See text for further explanations.
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